For one-dimensional (1D) substrates where \(d=1\) , solutions to Eq. (REF ) can not be derived from the exact results available for the full KPZ equation [1]}, [2]}, [3]}, [4]}, which require the Cole-Hopf transformation \(H(x,t)=\exp [\lambda h(x,t)/(2\nu )]\) . In higher dimensions, there are not exact analytical solutions even for the \(\nu \ne 0\)  KPZ equation. Hence, numerical approaches are necessary to assess the behavior of Eq. (REF ). Alternatively, discrete models in the same universality class could provide a description of the solutions of this equation.
